Intermittency in the small-time behavior of L\'evy processes

TL;DR

This paper investigates the small-time behavior of Lévy processes, providing precise asymptotics for moments and analyzing intermittency phenomena where moments decay at different rates.

Contribution

It offers a detailed analysis of moment convergence and intermittency in Lévy processes, extending understanding of their small-time asymptotics.

Findings

Precise asymptotics for all positive absolute moments.

Identification of critical moment order for convergence.

Demonstration of intermittency with varying decay rates.

Abstract

In this paper we consider convergence of moments in the small-time limit theorems for L\'evy processes. We provide precise asymptotics for all the absolute moments of positive order. The convergence of moments in limit theorems holds typically only up to some critical moment order and higher order moments decay at different rate. Such behavior is known as intermittency and has been encountered in some limit theorems.